$\text{Hom}(V,W)$ is the space of all linear transformations from $V$ to $W$. If $\dim V = m$ and $\dim W = n$, then $\dim\text{Hom}(V,W) = mn$. Here $m=3, n=4$, so $\dim\text{Hom}(\mathbb{R}^3,\mathbb{R}^4) = 3\times 4 = \mathbf{12}$.